Carrying Capacity of Spatially Distributed Metapopulations

The logistic equation, with carrying capacity, K, and growth rate, r, has traditionally been used to describe dynamics of ecological populations.Experiments confirmed the prediction that dispersal could increase metapopulation abundance in heterogeneous environments, whereas they rejected environments support a larger than homogeneous same sum over K values.Consumer-resource models, which explicitly consider resource inputs time scales feedbacks between organisms their resource, agree consistently experimental results, suggesting are more appropriate for describing populations space.The theoretical results have important management implications on wildlife, such as role dispersal, or habitat connectivity, influencing population patchy environments. Carrying capacity is key concept ecology. A body theory, based extended predictions spatially distributed, dispersing populations. However, this theory only recently tested empirically. disagree some when randomly two-patch system. consistent mechanistic model consumption an exploitable (consumer–resource model). We argue defined total equilibrium population, not fundamental property systems, at least context spatial heterogeneity. Instead, it emergent depends population’s intrinsic rates. (commonly upper limit size population), one most concepts ecology last century. As such, broadly used, from cell up communities landscape ecosystem levels [1.Chapman E.J. Byron C. flexible application ecology.Glob. Ecol. Consev. 2018; 13e00365Google Scholar,2.Del Monte-Luna P. et al.The ecosystems.Glob. Biogeogr. 2004; 13: 485-495Crossref Scopus (121) Google Scholar]. Wildlife biologists introduced term early 20th century tool wildlife management. Aldo Leopold viewed density reached particular site, determined by both resources available intraspecific competition [3.Sayre N.F. genesis, history, limits capacity.Ann. Assoc. Am. Geogr. 2008; 98: 120-134Crossref (148) Although, Leopold’s view, realized was usually less maximum under optimum conditions. He called saturation point – be achieved careful manipulation. definition no means held among ecologists. For instance, Paul Errington reach if there refuge predation available. Dhondt [4.Dhondt A.A. Carrying-capacity - confusing concept.Acta. Oecol-Oec. Gen. 1988; 9: 337-346Google Scholar] documented use Errington’s definitions other ecologists, noting, example, Dasmann [5.Dasmann R. Management. John Wiley & Sons, 1964Google carried distinctions further introducing four different related capacity: subsistence density, security tolerance density. reviewed multiplicity views confusing, concluding that, biology, should avoided. had already entered mainstream Odum [6.Odum E. Fundamentals Ecology. W. B. Saunders, 1953Google took first step giving formal mathematical meaning. constant Pearl–Verhulst form equation:dNdt=r1?NKN[1] where N r rate. 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Variation individuals fosters coexistence.Ecol. 21: 1496-1504Crossref (22) result, affecting crucial [35.Anderson K.E. al.Scaling responses variability advection-dominated systems.Ecol. 2005; 8: 933-943Crossref (40) Dispersal, process, plays shaping [36.Jønsson K.A. al.Tracking dispersal: individual community assembly dynamics.Trends 2016; 31: 204-214Abstract Scholar,37.Jacob S. al.Variability driver microcosms.Am. 194: 613-626Crossref (15) diversity composition [38.Albright M.B. Martiny J.B. Dispersal bacterial community.ISME 12: 296-299Crossref (41) [39.Little C.J. impact structured freshwater populations.Biol. 20180865Crossref (9) All organisms, including microbes, disperse within ranges occur ways. Two classic ideal free (IFD) random dispersal. According IFD, factors besides patch, predator ignored, animals move, continue move until cannot fitness. That attained equal capacities patches, integral whole area. In alternative movement, directional bias, symmetric diffusion special case proceeds steps. absence knowledge how actually preponderance models [40.Levin S.A. al.Theories simplification scaling distributed processes. Princeton Princeton, NJ1997: 271-295Google reaction–diffusion [41.Okubo A. Diffusion Ecological Problems: Mathematical Models. Springer, 1980Google Therefore, (Equation 1) complicates environment. Note, however, here asymmetric (Box 1).Box 1Mathematical Explanation Prediction 1Extending done reaction reaction-diffusion model, modeling collection discrete disperse. To demonstrate latter approach, simplified two each directions differ patches;dN1dt=r11?N1K1N1?DN1+DN2;[Ia] dN2dt=r21?N2K2N2?DN2+DN1.[Ib] D patches. found Freedman Waltman [87.Freedman H.I. Mathematical-models interactions .1. Stability 2 habitats without predator.SIAM Math. 1977; 631-648Crossref Holt [88.Holt R.D. Population environments: anomalous consequences distribution.Theor. Popul. 1985; 181-208Crossref (552) stated corrected Arditi [89.Arditi al.Asymmetric multi-patch equation.Theor. 120: 11-15Crossref (26) large rate (D ? ?), corresponds much faster change, isTotal population=K1+K2+K1?K2r1K2?r2K1r1K2+r2K1,[II] implication r1 r2, disperses symmetrically If K1 > K2 r1/K1 r2/K2, (equivalent diffusing) (Prediction 1). Conversely, < + K2. An analogous result continuous Lou [90.Lou Y. On migration single multiple species.J. Differ. Equations. 2006; 223: 400-426Crossref (171) partial differential equation. al. [68.Arditi al.Is beneficial capacity? insights 45-59Crossref (48) showed making does qualitatively. ?, simulations deviates smaller well, though approaching ?. influence environment.Thus, diffusing capacities, also differ. experimentally [47.Zhang al.Carrying connectivity.Ecol. 20: 1118-1128Crossref (52) Extending Thus, determining straightforward summation Ki, habitats. analyzed seemingly simple extension beyond nonspatial space, following surprising 1, Box 2), refer empirically.Box 2Mathematical 2Prediction follows II states but cases. is, case, = ?, ? deviation From thatTotal population=2K+2?Kr1?r2+?r1+r2Kr1+r2+?r1?r2,[I] exceeds 2K r2. mathematically correct, results. reason experiment differed way coupled equations. experiment, input resource/nutrient, provided yeast cases, amounts going patch case. shown I r2 violates equality cases.To this, note terms equations, r1N1? r2N2? r1K1 r2K2, seen adding Equations [Ia], [Ib] equilibrium. Now, comparing r1(K ?) r2(K r1K r2K, cases unless impossible exceed keeping constant. kept experiment. opposite different. (Theorem 11 [91.Guo Q. al.On environments.J. 81: 403-433Crossref (7) Scholar]; see [53.Wang Y.S. DeAngelis D.L. Energetic constraints paradox environment.Theor. 125: 30-37Crossref Scholar]). consumer–resource instead satisfied 3). system, spatially, nondispersing 2: environment,